Volume 9, 2003, Number 3

The squares of primes in the Modular Ring Z₄ fall in even rows, R₁, in Class ̅1₄ with R₁ = 6K_j and K_j = ½ n(3n ± 1) (depending on the parity of j). The n values are found to equal the rows that the primes occupy when Z₆ is set as a tabular array. The primes in Z₄ equal a unique sum or difference of squares (x² ± y²) and via n, the (x, y) pairs can be identified within the Z₆ structure. The n values for composites follow well-defined linear functions that permit easy sorting. Finally, the parameter n in the well-known function P = h2ⁿ − 1 has been identified as the row in one or more of the Modular Rings Z₁₀, Z₆ or Z₄ that contains one or more primes.

In [1] Krassimir Atanassov has introduced the arithmetic function RF(n) defined by RF(1) = 1 and RF(n) = q₁^α₁−1q₂^α₂−1… q_k^α_k−1 whenever n = q₁^α₁ q₂^α₂… q_k^α_k, where q₁, …, q_k are pairwise different prime numbers and we have α₁…α_k ≥ 1. The function RF(n) is called the restrictive factor. In the present paper we denote it simply by R(n).

A digital arithmetical function described in [1–3] will be defined and new its properties will be described.

The theory of Dirichlet series having number theoretic functions of a single variable as coefficients has a rich history. In this paper we present a parallel theory for iterated Dirichlet series with number theoretic functions of two variables as coefficients and find the Dirichlet product inverse of Ramanujan’s sum. The results presented here are easily accessible to an Advanced Calculus or undergraduate Number Theory course.